Step |
Hyp |
Ref |
Expression |
1 |
|
dia2dimlem7.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
dia2dimlem7.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
dia2dimlem7.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
4 |
|
dia2dimlem7.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
dia2dimlem7.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
6 |
|
dia2dimlem7.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
dia2dimlem7.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
dia2dimlem7.y |
⊢ 𝑌 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 ) |
9 |
|
dia2dimlem7.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑌 ) |
10 |
|
dia2dimlem7.pl |
⊢ ⊕ = ( LSSum ‘ 𝑌 ) |
11 |
|
dia2dimlem7.n |
⊢ 𝑁 = ( LSpan ‘ 𝑌 ) |
12 |
|
dia2dimlem7.i |
⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) |
13 |
|
dia2dimlem7.q |
⊢ 𝑄 = ( ( 𝑃 ∨ 𝑈 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ 𝑉 ) ) |
14 |
|
dia2dimlem7.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
15 |
|
dia2dimlem7.u |
⊢ ( 𝜑 → ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) |
16 |
|
dia2dimlem7.v |
⊢ ( 𝜑 → ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) |
17 |
|
dia2dimlem7.p |
⊢ ( 𝜑 → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
18 |
|
dia2dimlem7.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝑇 ) |
19 |
|
dia2dimlem7.rf |
⊢ ( 𝜑 → ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑈 ∨ 𝑉 ) ) |
20 |
|
dia2dimlem7.uv |
⊢ ( 𝜑 → 𝑈 ≠ 𝑉 ) |
21 |
|
dia2dimlem7.ru |
⊢ ( 𝜑 → ( 𝑅 ‘ 𝐹 ) ≠ 𝑈 ) |
22 |
|
dia2dimlem7.rv |
⊢ ( 𝜑 → ( 𝑅 ‘ 𝐹 ) ≠ 𝑉 ) |
23 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
24 |
23 1 4 5 6
|
ltrnideq |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐹 = ( I ↾ ( Base ‘ 𝐾 ) ) ↔ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) ) |
25 |
14 18 17 24
|
syl3anc |
⊢ ( 𝜑 → ( 𝐹 = ( I ↾ ( Base ‘ 𝐾 ) ) ↔ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) ) |
26 |
|
eqid |
⊢ ( 0g ‘ 𝑌 ) = ( 0g ‘ 𝑌 ) |
27 |
23 5 6 8 26
|
dva0g |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 0g ‘ 𝑌 ) = ( I ↾ ( Base ‘ 𝐾 ) ) ) |
28 |
14 27
|
syl |
⊢ ( 𝜑 → ( 0g ‘ 𝑌 ) = ( I ↾ ( Base ‘ 𝐾 ) ) ) |
29 |
5 8
|
dvalvec |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑌 ∈ LVec ) |
30 |
|
lveclmod |
⊢ ( 𝑌 ∈ LVec → 𝑌 ∈ LMod ) |
31 |
14 29 30
|
3syl |
⊢ ( 𝜑 → 𝑌 ∈ LMod ) |
32 |
15
|
simpld |
⊢ ( 𝜑 → 𝑈 ∈ 𝐴 ) |
33 |
23 4
|
atbase |
⊢ ( 𝑈 ∈ 𝐴 → 𝑈 ∈ ( Base ‘ 𝐾 ) ) |
34 |
32 33
|
syl |
⊢ ( 𝜑 → 𝑈 ∈ ( Base ‘ 𝐾 ) ) |
35 |
15
|
simprd |
⊢ ( 𝜑 → 𝑈 ≤ 𝑊 ) |
36 |
23 1 5 8 12 9
|
dialss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ ( Base ‘ 𝐾 ) ∧ 𝑈 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑈 ) ∈ 𝑆 ) |
37 |
14 34 35 36
|
syl12anc |
⊢ ( 𝜑 → ( 𝐼 ‘ 𝑈 ) ∈ 𝑆 ) |
38 |
16
|
simpld |
⊢ ( 𝜑 → 𝑉 ∈ 𝐴 ) |
39 |
23 4
|
atbase |
⊢ ( 𝑉 ∈ 𝐴 → 𝑉 ∈ ( Base ‘ 𝐾 ) ) |
40 |
38 39
|
syl |
⊢ ( 𝜑 → 𝑉 ∈ ( Base ‘ 𝐾 ) ) |
41 |
16
|
simprd |
⊢ ( 𝜑 → 𝑉 ≤ 𝑊 ) |
42 |
23 1 5 8 12 9
|
dialss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑉 ∈ ( Base ‘ 𝐾 ) ∧ 𝑉 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑉 ) ∈ 𝑆 ) |
43 |
14 40 41 42
|
syl12anc |
⊢ ( 𝜑 → ( 𝐼 ‘ 𝑉 ) ∈ 𝑆 ) |
44 |
9 10
|
lsmcl |
⊢ ( ( 𝑌 ∈ LMod ∧ ( 𝐼 ‘ 𝑈 ) ∈ 𝑆 ∧ ( 𝐼 ‘ 𝑉 ) ∈ 𝑆 ) → ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ∈ 𝑆 ) |
45 |
31 37 43 44
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ∈ 𝑆 ) |
46 |
26 9
|
lss0cl |
⊢ ( ( 𝑌 ∈ LMod ∧ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ∈ 𝑆 ) → ( 0g ‘ 𝑌 ) ∈ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) |
47 |
31 45 46
|
syl2anc |
⊢ ( 𝜑 → ( 0g ‘ 𝑌 ) ∈ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) |
48 |
28 47
|
eqeltrrd |
⊢ ( 𝜑 → ( I ↾ ( Base ‘ 𝐾 ) ) ∈ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) |
49 |
|
eleq1a |
⊢ ( ( I ↾ ( Base ‘ 𝐾 ) ) ∈ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) → ( 𝐹 = ( I ↾ ( Base ‘ 𝐾 ) ) → 𝐹 ∈ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) ) |
50 |
48 49
|
syl |
⊢ ( 𝜑 → ( 𝐹 = ( I ↾ ( Base ‘ 𝐾 ) ) → 𝐹 ∈ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) ) |
51 |
25 50
|
sylbird |
⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝑃 ) = 𝑃 → 𝐹 ∈ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) ) |
52 |
51
|
imp |
⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → 𝐹 ∈ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) |
53 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
54 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) |
55 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) |
56 |
17
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
57 |
18
|
anim1i |
⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝐹 ∈ 𝑇 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) |
58 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑈 ∨ 𝑉 ) ) |
59 |
20
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → 𝑈 ≠ 𝑉 ) |
60 |
21
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝑅 ‘ 𝐹 ) ≠ 𝑈 ) |
61 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝑅 ‘ 𝐹 ) ≠ 𝑉 ) |
62 |
1 2 3 4 5 6 7 8 9 10 11 12 13 53 54 55 56 57 58 59 60 61
|
dia2dimlem6 |
⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → 𝐹 ∈ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) |
63 |
52 62
|
pm2.61dane |
⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐼 ‘ 𝑈 ) ⊕ ( 𝐼 ‘ 𝑉 ) ) ) |